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Charged Eigenstate Thermalization, Euclidean Wormholes and Global Symmetries in Quantum Gravity
by Alexandre Belin, Jan de Boer, Pranjal Nayak, Julian Sonner
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Submission summary
Authors (as registered SciPost users):  Pranjal Nayak · Julian Sonner 
Submission information  

Preprint Link:  scipost_202106_00040v1 (pdf) 
Date submitted:  20210622 18:45 
Submitted by:  Nayak, Pranjal 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We generalize the eigenstate thermalization hypothesis to systems with global symmetries. We present two versions, one with microscopic charge conservation and one with exponentially suppressed violations. They agree for correlation functions of simple operators, but differ in the variance of charged onepoint functions at finite temperature. We then apply these ideas to holography and to gravitational lowenergy effective theories with a global symmetry. We show that Euclidean wormholes predict a nonzero variance for charged onepoint functions, which is incompatible with microscopic charge conservation. This implies that global symmetries in quantum gravity must either be gauged or explicitly broken by nonperturbative effects.
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Reports on this Submission
Report 2 by Daniel Harlow on 202196 (Invited Report)
 Cite as: Daniel Harlow, Report on arXiv:scipost_202106_00040v1, delivered 20210906, doi: 10.21468/SciPost.Report.3498
Report
This is a nice paper on Euclidean wormholes and global symmetries in quantum gravity, as well as the generalization of the Eigenstate Thermalization Hypothesis (ETH) to the case of a $U(1)$ global symmetry. There are several points where I think improvement is needed, two major and a few more minor. First the two major points:
1) The discussion of bulk gauge symmetries around figure 1 is flawed. In particular, contrary to the text and the caption of figure 1, it is not necessary for the two operators to be connected by a Wilson line. Each can instead be connected to its respective boundary. And in fact this is implicitly assumed in the text above equation 12, since it is only if we attach them to the boundaries that we get something which transforms nontrivially under $U(1)\times U(1)$. The situation with the Wilson line connecting them as shown in figure 1 is actually the correlation function $\langle \phi(x)^\dagger W(x,y)\phi(y)\rangle$ which is discussed below equation 12, and that correlator does not need to vanish even if the operators are extrapolated to the boundaries (even after extrapolating them the operator $\phi(x)^\dagger W(x,y)\phi(y)$ is still neutral under $U(1)\times U(1)$ since the transformation of the Wilson line cancels the transformations of the scalar operators.
2) The idea that Euclidean wormholes violate global symmetries has a long history, and one which is not mentioned at all in this paper. This is a referencing error that needs to be fixed. Just to give a sampling,
https://www.sciencedirect.com/science/article/abs/pii/0550321388901095?via%3Dihub
https://www.sciencedirect.com/science/article/abs/pii/0550321389905038?via%3Dihub
https://www.sciencedirect.com/science/article/abs/pii/0550321390901498?via%3Dihub
https://arxiv.org/pdf/hepth/9502069.pdf
In addition I have some further minor comments/questions:
3) I think the notation of equations 38 would be greatly improved by labeling the states as $i\rangle$, where $i$ runs over a basis of simultaneous eigenstates of $H$ and $Q$, instead of $E_iQ_i\rangle$. The latter makes it look like we can have states with different $E$ and $Q$ for the same $i$, which is not correct. It is especially confusing e.g. in equation 5 where a sum is written over $E_j$ and $Q_j$: this should really just be a sum over $j$!
4) Below equation (3) it is said that $f_a$ and $g_a$ are smooth functions of $Q_i$. But what does it mean to be a smooth function of a discrete variable? I sort of know what they mean, but the authors are really trying to nail down a precise version of ETH with a $U(1)$ symmetry it might be worth spelling this out more.
5) Is equation (4) supposed to hold for arbitrary $q$, or only for $q$ which is small in some sense? In particular ETH is supposed to be closed under products of operators, but that doesn't seem to be the case here due to the somewhat arbitrary choice of taking the average of the entropy at the two different charges unless we make some kind of assumption about the smallness of $q$.
6) A fair number of the references are missing arxiv/journal information, these should be checked.
Anonymous Report 1 on 202195 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202106_00040v1, delivered 20210905, doi: 10.21468/SciPost.Report.3471
Strengths
1. Well embedded into prior literature.
2. Clear motivations.
3. Clear discussions and future directions.
Weaknesses
1. Technical aspects are very brief. They assume the reader is very familiar with results in prior work, and omit various definitions.
2. The manuscript would have benefited from exploring the future directions discussed. It relies on one very simple observation about global symmetries in ETH, and its interpretation in AdS/CFT.
Report
This manuscript proposes a version of the charged ETH, and what that hypothesis implies on quantum gravity via the AdS/CFT correspondence. This leads them to arguments against global symmetries in quantum gravity.
It is a wellwritten manuscript, but its short format obscures some aspects of the results presented. It would benefit from expanding on certain portions to help place the results on a more firm footing.
Requested changes
Since the authors are not restricted to formatting and lengths of other journals, my requested changes are basically directed to expand the text.
1. Please change the formatting of the references. Add appropriate arxiv hyperlinks.
2. More explanations for eqn (11) are needed. What is $h(s)$? I assumed it is the action of an element of $\Gamma$ acting on geodesic length. Also, what is the symbol $\sim$ indicating here?
3. Below eqn (12), $W(x,y)$ is not defined. I assume it refers to the Wilson line of the $U(1)$ gauge field between the two points. Also, add a reference to support the claim that $\langle\phi(x)^\dagger W(x,y)\phi(y)\rangle$ does not vanish if the points don't reach the boundary.
4. I'm not sure I agree with the sentence "Here, this leads to a paradox." on page 5. It is not really a paradox: as interpreted and used as in the outcome here, the tension between eqn (11) and (14) leads to a constraint that makes predictions but it is not paradoxical. I would invite the authors to reconsider the opening of that paragraph.
5. It is strange that eqn (2) is stated as a summary of results in the introduction, but never directly referred to in the main sections. This is tied to the last paragraph of section 4 (top right column page 5) being rather vague, where I suspect quoting eqn (2) would have been appropriate. Maybe the authors can connect better the content of the summary with the main text.
6. Are there any examples of Hamiltonians, or quantum systems, that one can verify that they comply with eqn (8) explicitly? Should one think about this as a spontaneous or explicit symmetrybreaking mechanism? It is not very clear what the terms "approximate global symmetry" or "mildly break charge conservation" mean in practice.